-
Photonics and Nanostructures – Fundamentals and Applications 12
(2014) 376–386
Available online at www.sciencedirect.com
ScienceDirect
THz metamaterials made of phonon-polariton materials
M. Kafesaki a,b,∗, A.A. Basharin a, E.N. Economou a,c, C.M.
Soukoulis a,da Institute of Electronic Structure and Laser,
Foundation for Research and Technology, Hellas (FORTH), P.O. Box
1385, Heraklion, Crete, Greece
b Department of Materials Science and Technology, University of
Crete, Greecec Physics Department, University of Crete, Greece
d Ames Lab and Department of Physics and Astronomy, Iowa State
University, Ames, IA, USA
Received 24 March 2014; received in revised form 14 May 2014;
accepted 28 May 2014Available online 10 June 2014
Abstract
In this paper, we demonstrate numerically various phenomena and
possibilities that can be realized in THz metamaterials madeof
phonon-polariton materials. Such phenomena include hyperbolic
dispersion relation, subwavelength imaging using backward
propagation and backward radiation, total transmission and
subwavelength guiding exploiting Mie-resonant scattering in
permittivitynear zero host, and toroidal dipolar response. The
systems that we use to demonstrate most of these phenomena are
two-dimensionalperiodic systems of �m-scale rods in a host, where
both rods and host are made of polaritonic alkali-halide
materials.© 2014 Elsevier B.V. All rights reserved.
1. Introduction
The continuously expanding technologies for THzsources along
with the unique properties and capabilitiesof the THz radiation,
especially in the security, sensingand communications domains,
constantly increasenowadays the demand for the realization of
componentsfor the manipulation of THz waves [1,2].
Since the majority of natural materials do not showstrong
response to the THz waves, and so they do notoffer themselves for
THz handling and a straightforwardTHz component realization,
metamaterials (i.e. artifi-
cial materials, structured in subwavelength scales andshowing
novel and unique properties, unattainable innatural materials), can
offer an excellent solution to the
∗ Corresponding author at: Foundation for Research and
Technology,Hellas (FORTH), P.O. Box 1385, Heraklion, Crete,
Greece.Tel.: +30 2810 391547; fax: +30 2810 391569.
E-mail address: [email protected] (M. Kafesaki).
http://dx.doi.org/10.1016/j.photonics.2014.05.0091569-4410/©
2014 Elsevier B.V. All rights reserved.
problem. This is because the metamaterials response isbased
mainly on geometry-induced resonances [3] whichcan be easily tuned
in frequency by adjusting the size oftheir basic building blocks.
Moreover, the richness in thephenomena and properties achievable
with metamateri-als (like, e.g. very large, near zero or negative
permittivityand/or permeability [3], negative refractive index
[4,5],giant chirality [6,7] etc.) leads to a variety of
capabilitiesfor the metamaterial-based systems [8], including
perfectabsorption [9,10], subwavelength resolution imaging[11],
polarization filtering and manipulation [12], spa-tial and temporal
filtering, etc., which can be greatlyexploited in the THz
domain.
The majority of today’s THz metamaterials usemetal as the basic
material for the metamaterials ele-ments. There, the desired
metamaterial property is
achieved by properly designed current resonances inthe
subwavelength-size metallic elements (known asmeta-atoms). An
alternative option to achieve useful andinteresting metamaterial
properties is to use meta-atoms
dx.doi.org/10.1016/j.photonics.2014.05.009http://crossmark.crossref.org/dialog/?doi=10.1016/j.photonics.2014.05.009&domain=pdfhttp://www.sciencedirect.com/science/journal/00000000mailto:[email protected]/10.1016/j.photonics.2014.05.009
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M. Kafesaki et al. / Photonics and Nanostructure
ade of a high-index dielectric, instead of a metal, andxploit
the Mie-resonances of the meta-atom [13,14].ue to the high-index of
the meta-atom these resonances
ppear at wavelengths much larger than the meta-atomize, allowing
thus the treatment of the total metamate-ial as a homogeneous
material with effective properties.
category of materials which in the THz regime com-ines both the
properties and capabilities of metals and ofigh index dielectrics,
and therefore provides an excel-ent candidate for THz metamaterial
building blocks,s polaritonic materials [15]. Polaritonic materials
areolar crystals where the incident radiation excites crys-al
vibrations (transverse optical phonons) in the crystal.he coupling
of the incident radiation with the inducedeld created by the
crystal vibrations results to the so-alled phonon-polariton modes,
and is described by aesonant permittivity response of Lorenz
type:
¯ = ε∞(
1 − ω2L − ω2T
ω2 − ω2T + iωγ
)(1)
In Eq. (1) ωT is the angular frequency of the trans-erse optical
phonons in the crystal, which falls in theHz regime and at which
the permittivity blows up (in
he absence of losses), ωL is the angular frequency of
theongitudinal optical phonons in the crystal (ωL > ωT), and∞ is
the limiting value of the permittivity for high fre-uencies, well
above ωL but well below the electronicap. The values of ωT, ωL and
ε∞ are related to thearameters of the polar crystal.
The dielectric function (1), in the frequency regimeust above
the resonance frequency, ωT, shows nega-ive values, similar to that
of the dielectric function of
etals in the optical regime (see, e.g. Fig. 1). This indi-ates
that all the effects that can be observed in metalptics
(plasmonics, optical metamaterials) can be trans-erred to the THz
using polaritonic materials [16–19],omething that offers to the
polaritonic materials uniqueower for THz wave manipulation.
Moreover, the highositive values of the dielectric function just
below theesonance frequency, reaching in some cases values evenf
the order of few hundreds (e.g. for LiTaO3 and TlCl)20], can be
exploited in a variety of high-index-basedetamaterial phenomena,
such as negative effective per-eability and/or permittivity,
negative refractive index,
tc. [14,20–24]. Another particularly interesting fre-uency
region encountered in polaritonic materials ishere the dielectric
function gets values close to zero,
.e. for frequencies around the frequency of longitu-inal optical
phonons, ωL. The electromagnetic (EM)esponse of materials with
epsilon near zero (ENZ) haseen studied extensively recently, mainly
in connection
damentals and Applications 12 (2014) 376–386 377
to metamaterials (since metamaterials can easily allowsuch kind
of response), and interesting phenomena andpossibilities have been
proposed and demonstrated. Suchphenomena include possibility to
squeeze EM wavesat will using narrow ENZ channels [25,26],
possibilityto easily shape the radiation pattern of sources
embed-ded or in close proximity to ENZ structures [27],
stronginteraction of phonons with charge carriers resulting
toinstabilities generating THz waves [28,29], and others[30].
This broad range of permittivity values achievablein polaritonic
materials is of great importance if suchmaterials are properly
shaped, providing metamate-rial elements. Combining the rich
shaping possibilitiesachievable by the modern fabrication
approaches withthe wide range of permittivity values achievable
inpolaritonic materials, one can result in a uniquelyrich variety
of phenomena which can have a greatimpact in all the modern THz
applications. In this workwe demonstrate numerically few of such
phenomena,i.e. phenomena observable in metamaterials madeof
phonon-polariton materials, called here polari-tonic metamaterials.
In particular, we demonstrate: (a)hyperbolic dispersion relation
response in anisotropicpolaritonic metamaterials (see Section 3),
leading tosubwavelength resolution imaging; (b)
subwavelengthimaging and guiding due to backward radiation in
polari-tonic waveguides (see Section 4); (c) total transmissionand
extremely subwavelength guiding in metamaterialsmade of dielectric
cylinders in an ENZ polaritonic host(see Section 5); (d) toroidal
response in periodic sys-tems built by proper clusters of
polaritonic cylinders(as building blocks), due to the high epsilon
responseof the cylinders (see Section 6). The basic systems
andmaterials that we use to demonstrate the above men-tioned
phenomena, as well as the simulation approachesemployed, are
discussed in the next section.
2. Basic systems and their electromagneticproperties
The systems that we will employ to demonstratemost of the
polaritonic metamaterial based phenomenadiscussed here are
alkali-halide polaritonic systems com-posed of LiF rods embedded in
either NaCl or KCl host,in hexagonal periodic arrangement [31]. A
drawing ofsuch a system is shown in Fig. 1(a), while Fig. 1(b)shows
the relative dielectric function of LiF and NaCl
[16]. The dielectric function of KCl is very similar tothat of
NaCl, with the polaritonic resonance appearingat around 4.2 THz
[16]. An important merit of those sys-tems is that they can be
easily fabricated in micro-meter
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378 M. Kafesaki et al. / Photonics and Nanostructures –
Fundamentals and Applications 12 (2014) 376–386
itonic h
Fig. 1. (a) The geometry of the polaritonic rods embedded in a
polarrelative permittivity of LiF and NaCl.Source: Data are taken
by Palik [16].
length scales using self-organization of eutectic mix-tures
[32,33]. With this approach one can easily adjustthe length-scale
of the fabricated structures during thefabrication process,
obtaining structures with differentlattice-sizes and rod-diameters.
The particular systemsthat we analyze here have been already
fabricated, in dif-ferent length scales, and their electromagnetic
responsehas been experimentally studied using reflection
and/ortransmission measurements [31,34]. The
geometricalcharacteristics of the fabricated systems are listed
inTable 1.
The electromagnetic response of the fabricated struc-tures has
been studied also numerically and has beenanalyzed using effective
medium formulas. Comparisonwith the experimental data showed that
the responseof the smaller of the systems can be reproduced
andexplained using the Maxwell-Garnett effective mediumapproach
[35], summarized by Eq. (2) below, which is
valid in the quasistatic regime, where the wavelength inboth the
rods and host is larger than the rod size and the
Table 1Geometrical characteristics of the fabricated
alkali-halide systems thathave been studied in this work. The rod
diameters and lattice constantsare measured average values. The
structures exhibit deviations fromthe perfect regularity. For
details see Ref. [31].
System LiF volumefraction
Rod diameter(Lattice constant)[�m]
LiF rods in NaCl 25% 2.0 (3.6) to 10.7(20.3)
LiF rods in KCl 6.9% 0.8 (2.8) to 6.4(23.3)
ost system and the polarizations of the incoming EM wave. (b)
The
lattice constant:
ε⊥eff = εhost(1 + ϕ)εrods + (1 − ϕ)εhost(1 − ϕ)εrods + (1 +
ϕ)εhost ,
ε‖eff = ϕ εrods + (1 − ϕ)εhost . (2)
In Eq. (2) ϕ is the rod filling ratio and the super-scripts
parallel and perpendicular refer to the electricfield polarization
relative to the rods axes – see Fig. 1(a).Regarding the
larger-scale systems, they can also bedescribed as effective media
[36], but there moreelaborated approaches should be employed, such
asPendry’s averaging field approach [13,37],
extendedMaxwell-Garnett approach [38,39] and others [40]. Thisis
due to the high-index response of the rods belowthe frequency ωT
(see Eq. (1)), which results to veryshort wavelengths inside the
rods and thus to sub-wavelength Mie-resonances for even relatively
smallrod-diameters (note that the Mie-resonances of the rodsappear
when the wavelength inside the rods is compa-rable to the rod
diameter) [14,41]. The requirement ofeffective medium approaches
going beyond the quasi-static Maxwel-Garnett approach is quite
common in theeffective medium description of media composed of
highindex dielectrics.
In the discussion below we focus in most of the caseson the
smallest among the fabricated structures men-tioned in Table 1, and
when effective medium descriptionis required we employ the
Maxwel-Garnett approach.For achieving detailed numerical data we
employed the
Microwave Studio commercial software, where the com-ponent
material permittivity has been entered via theLorenz model (Eq.
(1)) with characteristic parameters asmentioned in Table 2. We have
to mention here that the
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M. Kafesaki et al. / Photonics and Nanostructures – Fundamentals
and Applications 12 (2014) 376–386 379
Table 2The characteristic frequencies appearing in the Lorenz
(Eq. (1)) description of the component materials of the structures
discussed here (ε0 is thepermittivity of the vacuum).
Material Resonance frequencyfT =ωT/2π [THz]
Longitudinal phononfrequency fL = ωL/2π[THz]
Damping frequencyγ’ = γ/2π [THz]
Asymptotic relativepermittivity ε∞/ε0
LiF 9.22 19.1 0.527 2.027NaCl 4.92 7.8 0.207 2.222K
pwω
trdd
3p
arsltlaogatt(smcptrlpHi
stt(t
Cl 4.21 6.2
arameters of Table 2 for the LiF, while approaching veryell the
material response close to resonance frequencyT show slight
deviation from the experimental permit-
ivity values [16] around the frequency ωL. This mayesult in
small deviations in frequency of the phenomenaiscussed here if
compared with future experimentalata.
. Hyperbolic dispersion relation in THzolaritonic
metamaterials
Structures with hyperbolic dispersion relation, knowns
hyperbolic metamaterials [42,43], have been provedecently as a very
important category of metamaterials,ince they show the ability for
subwavelength reso-ution imaging (known as hyperlensing [44])
withouthe need for resonant response (which implies highosses) or
magnetic response (which is not easy to bechieved, especially in
high frequencies), while theyffer an easy practical realization due
to the simpleeometries involved. Such metamaterials are
uniaxialnisotropic structures with one principal component ofhe
effective permittivity (or permeability) negative andhe others
positive, resulting to equifrequency surfacesi.e. surfaces of
constant frequency in k-space) of thehape of hyperbola (unlike the
ellipsoid-like surfaces ofetamaterials with all the permittivity
(or permeability)
omponents being of the same sign) [45]. The easierractical
realization of such metamaterials is encoun-ered in systems of
negative permittivity (e.g. metallic)ods periodically placed in a
dielectric host, or in lamel-ar (multilayer) systems alternating
negative and positiveermittivity layers (i.e. metal-dielectric
layers) [43].ere we demonstrate hyperbolic metamaterial
response
n polaritonic systems.The ability of hyperbolic metamaterials
(HMM) for
ubwavelength resolution imaging is based on the fact
he HMM do not have an upper limit in the magni-ude of
propagating wave vectors that they can supportdue to the unbounded
form of the hyperbola, in contrasto ellipsoid or sphere), so they
can transmit arbitrarily
0.156 2.045
large (in principle) wave vectors [44,46]. In a regularmedium,
wavevector components larger than ωn/c (n isthe refractive index of
a medium and c the vacuum wavevelocity) along one direction
correspond to evanescentwaves along the perpendicular direction and
are lost ifpropagation occurs in regular media. These
componentsthough, which carry the finest details of the source
objectemitting them (forming the basis of Pendry’s super-lens
[11]), can be coupled to propagating waves in aHMM and transmitted
without attenuation (beyond theattenuation associated with the
losses of the materials)[46,47]. Besides the subwavelength
resolution imaging,the hyperbolic dispersion is associated also
with broad-band high density of EM modes, which can greatly
affectthe performance of EM sources placed in HMM [48–50].Other
interesting possibilities that can also be realized inHMMs are
advanced absorption and thermal emissioncontrol [51,52], cloaking
possibilities [53], possibilitiesto mimic phenomena related to
gravitation theory [54]etc.
HMM have been realized so far in the IR and opti-cal regime
using periodic systems of metallic rods ina host [55–57] or
metallic layered systems [58,59],and subwavelength resolution
imaging has been demon-strated, even associated with image
magnification whenthe layers are properly curved [44] (for a review
seeRef. [43]). Here we show that the same effects canbe observed in
the THz regime in systems of polari-tonic rods in a host,
exploiting the negative permittivityresponse of the rods [31,36].
Such a system is the LiFrods in NaCl host system shown in Fig.
1(a), in the fre-quency region between 9.5 and ∼13 THz. Applying
theMaxwell-Garnet effective medium formula (Eq. (2)) forsuch a
system, which, as we mentioned earlier, is valid forrod diameters
smaller than 2 microns, one achieves theeffective permittivity
response shown in Fig. 2. The per-mittivity of Fig. 2 for the
frequency region 9.5–13 THz
shows negative values for electric field (E) parallel tothe rods
and positive for E perpendicular to the rods (seeFig. 1(a) for the
description of polarization), suggestinghyperbolic metamaterial
response. Selecting an interface
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380 M. Kafesaki et al. / Photonics and Nanostructures – Fun
Fig. 2. Effective relative permittivity for a system of parallel
infinitelylong LiF rods embedded in NaCl, for LiF volume fraction
25%, and forthe two different possible polarizations of an incident
EM field relativeto the rod axes (see Fig. 1(a)). The effective
permittivity was calculatedusing the Maxwel-Garnett effective
medium formulas (Eq. (2)), where
for the permittivity of the component materials the Lorenz
approxima-tion of the data presented in Fig. 1(b) was used, with
parameters thoselisted in Table 2.
perpendicular to the rods (see Fig. 3(b)) and calculatingthe
dispersion curves kparallel vs kperpendicuar, where theparallel and
perpendicular here refer to the interface, oneachieves the curves
shown in Fig. 3(a), demonstrating thehyperbolic dispersion relation
response.
This hyperbolic dispersion relation and the resultingpossibility
for subwavelength resolution imaging in our
LiF in NaCl system was demonstrated also in detailednumerical
simulations. An example is shown in Fig. 3(b),where a localized
source emitting a TM wave is placedclose to the interface between
air and the LiF–NaCl sys-
Fig. 3. (a) Dispersion curves for the extraordinary wave
propagating alongdemonstration of superlensing in that metamaterial
by showing the magnetik-component parallel to the interface between
air and the metamaterial, i.e. alincident wave in panel (b) is a TM
wave (magnetic field along y-direction andof the LiF rods in the
system is 2 �m and the lattice constant of the
hexagonatwo-dimensional cross-section of the system presented in
panel (b) we have m
damentals and Applications 12 (2014) 376–386
tem, and imaging with resolution ∼�/4 is demonstratedfor
frequency 10.86 THz. Note that the system lengthhere is chosen as
to match the effective wavelength inthe system, exploiting thus the
Fabry-Perot resonancesof the system, which allow maximum
transmission.
One has to point here that the employment ofpolaritonic
materials as a means to achieve hyperbolicdispersion relation
response is associated with certainadvantages, compared to
corresponding metallic sys-tems: (a) In polaritonic systems one
avoids the problemof spatial dispersion which is common in metallic
struc-tures operating in microwaves and THz and complicatesthe
structure response [60,61]. (b) The loss factor,Im(ε)/Re(ε), of
polaritonic materials in the frequencyregion of hyperbolic
dispersion is smaller (it is of theorder of 10) than that of metals
in the optical region(where the permittivity values are similar),
allowinglarger propagation distances in THz polaritonic
metama-terials than in similar nanoscale metallic systems.
This,combined with the much easier experimental realiza-tion of the
THz polaritonic structures, shows that suchstructures offer an
ideal system for demonstration andanalysis of the effects and
possibilities associated withhyperbolic dispersion relation.
4. THz superlensing based on backwardradiation in negative
permittivity waveguides
Waveguides made of polaritonic materials possess
some unique properties unattainable by other materialsin the THz
frequency range, due to the small negativepermittivity values that
polaritonic materials possess asthe frequency approaches ωL. In
waveguides with small
the rods in the LiF in NaCl metamaterial studied (see Fig.
1(a)); (b)c field component Hy propagating along the structure.
kparallel is theong x-direction of panel (b), and kperpendicuar is
along z-direction. Theelectric field in the x–z plane) of frequency
10.86 THz. The diameterl lattice 10.7 �m. To distinguish between
rods and host stripes in thearked the rod position by white
dots.
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M. Kafesaki et al. / Photonics and Nanostructures – Fundamentals
and Applications 12 (2014) 376–386 381
F pagated requencl
nbaot
w(stfiria
k
ntc2tk“cibdbvbtla
two-dimensional cross-section of the system presentedin panel
(b) we have marked the rod position by whitedots.
Fig. 5. Subwavelength propagation and imaging due to backward
radi-ation in the system of LiF rods in NaCl that was discussed in
SectionIII and Fig. 3, in the frequency region of small negative
LiF permit-tivity (here 16.24 THz). The propagation along
z-direction in the LiFrods is backward leading to convergence of
the leaked radiation in
ig. 4. (a) Scheme of the focusing mechanism based on backward
proistribution in a LiF slab of thickness d = 2 �m embedded in NaCl
at focated 1 �m away from the LiF–NaCl interface.
egative permittivity values there is the possibility ofackward
guided modes (i.e. modes of opposite phasend group velocity) which
show quite strong leakageutside the guide, leading to backward
radiation fromhe side-walls of the guide [62].
To understand this better let’s consider a planar slabaveguide
of relative permittivity ε1 and thickness d
along x-direction – infinite along y and z directions –ee Fig.
4(a)) embedded in a material of relative permit-ivity ε2. For TM
electromagnetic modes (i.e. magneticeld in the plane of the slab
boundaries) the dispersionelation of the even guided modes (i.e.
modes propagat-ng along the guide and evanescent outside the
guide,long x-direction) [62] is
2xd = k1xd ε2ε1
tan(k1xd), with
k1x =√
k20ε1 − q2, k2x =√
q2 − k20ε2. (3)
In Eq. (3), k0 = ω/c, q is the wave-vector compo-ent along the
waveguide, which is continuous acrosshe guide boundaries, and the
subscripts in the k-omponents denote the medium (1 for the
waveguide,for the surrounding medium) and the direction. Using
he above equations to solve the dispersion relation q vs0, one
can find for ε1 < 0 the existence of backwardguided” waves,
which have also a large attenuationomponent (Im(q)) along the
direction of the guide,ndicating leakage outside the guide across
the guideoundaries. For large negative values of ε1 these waveso
not dominate the flux propagating along the guide,ut under certain
conditions, e.g. for small negativealues of ε1 (as compared in
magnitude to ε2) these
ackward waves dominate the guided flux and, dueo the
conservation of q, lead to negatively refractedeaking waves, i.e.
backward radiation outside the guidelong the x-direction. Thus if
one places a point source
d and radiating guided modes; (b) magnetic field component Hy
fieldy 13.5 THz. The source is a TM point source (magnetic field
along y)
outside a waveguide having small negative permittiv-ity values,
as to excite the backward guided modes, thenegatively refracted
leaking waves converge at the otherside of the guide (see Fig. 4)
resulting to an image ofthe point source; this makes the guide
operating as aflat lens, providing also subwavelength resolution
due tothe restoration of the evanescent waves emitted by
thesource.
Exploiting further the backward radiating modesmentioned above,
one can achieve subwavelength imag-ing (or subwavelenth tunneling
of electromagneticwaves) in a system of parallel cylindrical
waveguides,such as the LiF rods (acting as waveguides) in NaClhost
system, which was discussed in the previous sec-tion, in the
frequency regime around 16 THz where LiFhas small negative epsilon
values and NaCl behaves as adielctric [16]. Such a subwavelength
imaging is numer-ically demonstrated in Fig. 5.
To distinguish between rods and host stripes in the
the NaCl host due to negative refraction. The point source emits
a TMwave (magnetic field along y-direction), and figure shows the
magneticfield component Hy . The white dots in the figure are used
to mark theposition of the rods, as described also in Fig. 3.
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382 M. Kafesaki et al. / Photonics and Nanostructures –
Fundamentals and Applications 12 (2014) 376–386
Fig. 6. (a) The scattered electric field by a dielectric
cylinder (of relative permittivity 40 and radius 4 �m) embedded in
an ENZ medium at the regionwave isembeddidentica
a narrow subwavelength channel where the wave prop-agates by
being transferred from scatterer to scatterer,
Fig. 7. Subwavelength guiding exploiting the Mie-resonances of
LiFrods in NaCl, at frequency 7.85 THz where NaCl has permittivity
near
of a cylinder resonance (4.54 THz). The electric field in the
incomingalong an array of dielectric cylinders (as the one shown in
panel (a))scattered field. The guiding occurs at frequency 4.55
THz, i.e. almost
5. Epsilon near zero based total transmission andsubwavelength
guiding in polaritonic systems
As was mentioned in the introduction, one can achievea variety
of interesting phenomena in media with epsilonnear zero; this is
mainly due to propagation with practi-cally no phase advance that
can take place in such media[26,63,64]. Even more interesting and
rich phenomenacan be observed if one combines properly
structuredENZ media with normal dielectrics [65,66]. For exam-ple,
inserting regular dielectric scatterers in an ENZmedium one can
observe similar phenomena to the onesachieved in media composed of
high-index dielectricscatterers in a regular dielectric host, such
as negativepermeability and/or permittivity. This is due to the
largewavelength in the ENZ host (allowing the
scatterer’sMie-resonances to occur in subwavelength regions)and the
too high contrast in the dielectric parametersbetween scatterers
and the ENZ host. Other interest-ing phenomena occurring in media
with scatterers in anENZ host are total transmission or total
reflection or EMcloaking for TM incident wave polarization if the
scatter-ers are made of a PEC material [67], and, as was
shownrecently, total transmission and extreme subwavelengthguiding
of TE polarized waves in media with cylindricaldielectric scatteres
in an ENZ host [30,34].
The presence of epsilon near zero frequency regionsin
polaritonic materials, combined with the easiness inthe fabrication
of composite media made of polaritonicmaterials, allows the easy
observation and exploitation inTHz of all the interesting phenomena
that can observed incomposite media of scatteres in ENZ host. We
showedrecently that such interesting phenomena are extreme
subwavelength guiding associated with total EM wavetransmission
[30,34].
Analyzing the propagation of a plane electromagneticwave coming
from vacuum and impinging on an ENZ
parallel to the cylinder. (b) Very confined almost perfect
propagationed in an ENZ medium due to the p-like symmetry of each
individuall to the frequency of a Mie-resonance.
medium with embedded dielectric cylinders [30], for Eparallel to
the cylinders one can observe extremely sub-wavelength and strongly
confined Mie-resonances in thecylinders, where the electric field
distribution shows thesymmetry of p-like atomic orbitals oriented
along thedirection of the incident wave (see Fig. 6(a)). It can
beshown both analytically and numerically [30] that thepresence of
such resonances: (a) maximizes the couplingof the resonators along
propagation direction if cylindri-cal resonators are placed next to
each other, and (b) canresult in almost total transmission of the
incident waveby the system.
These effects can be exploited in an ordered lineararrangement
of dielectric cylinders placed in an ENZhost, as shown in Fig.
6(b). Such an arrangement acts as
zero and LiF relative permittivity 24.8. The guiding is
demonstrated byshowing the electric field component Ez of an
incident wave comingfrom a slit on PEC screen. The rod diameter
here is 6.4 �m and thelattice constants are 12 �m along x-direction
and 15 �m along y.
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M. Kafesaki et al. / Photonics and Nanostructures – Fundamentals
and Applications 12 (2014) 376–386 383
Fig. 8. (a) A dielectric metamolecule supporting toroidal
dipolar response for the incident electromagnetic field
configuration shown in the figure.The metamolecule is a cluster of
4 infinitely long cylinders of radius R = 8 �m and center-to-center
separation a = 18 �m. The metamaterial is formedb d d = 58fi at
frequ
fo
EoEtdfia7wEotstcE
6
irbSFms
y placing the metamolecules periodically along the x-axis with
perioeld induced in the metamolecule of panel (a), made by LiTaO3
rods
orming an extremely subwavelength and efficient (i.e.f high
transmission) waveguide.
Subwavelength guiding due to Mie-resonances inNZ host can be
sustained even in two-dimensional peri-dic systems, such as the LiF
rods in NaCl host at theNZ response region of the host, since the
coupling of
he neighboring cylinders along propagation directionominates
(due to the directional form of the resonantelds). Fig. 7
demonstrates this response for a 2D rect-ngular periodic system of
LiF rods in NaCl, at frequency.85 THz where NaCl behaves as an ENZ
material. Sub-avelength guiding with enhanced transmission due toNZ
host has been also demonstrated in periodic systemsf LiF rods in
KCl as the ones described in Table 1; there,he transmission, which
was studied also experimentally,howed a peak at the ENZ region of
KCl (despite the largehickness of the system and the non-perfect
regularity),onfirming the possibility for high transmission due
toNZ host response [34].
. Toroidal response in eutectic metamaterials
Besides metamaterials made of periodically placedndividual rods,
metamaterials consisting of clusters ofods, or more generally
clusters of scatterers, as basicuilding blocks can also provide
very interesting effects.
uch effects that have been already demonstrated areano
resonances in plasmonic clusters and in clustersade of high index
dielectric particles [68,69], ultra-
trong electromagnetic field localization inside clusters
�m. (b) Calculated distribution of the absolute value of the
magneticency 1.89 THz. The arrows show the magnetic flux.
of particles, leading to enhancement of non-linearities[70],
etc.
As it has been shown recently, a particularly inter-esting
effect that becomes possible in properly shapedclusters of high
index dielectric cylinders (see Fig. 8)is toroidal dipolar response
[71–73]. Toroidal dipolarresponse is created by currents
circulating on a surface ofa torus perpendicular to its axis,
resulting in a magneticflux having the shape of a torus [74]. Such
a response,although it results from an elementary current
excitation,is usually omitted from the standard multipole
analysis;it is associated though with a multitude of unusual
andinteresting phenomena, such as non-reciprocal refrac-tion of
light [75], generation of vector potentials withoutpresence of
electric fields [76], etc.
In the context of metamaterials toroidal responsehas been
demonstrated so far by using clusters ofmetallic elements, such
like split-ring resonator clus-ters [74,77–79]. Only recently the
realization of suchresponse in high index dielectrics was
demonstrated[71], specifically in systems composed of
torus-like-shaped clusters of high index dielectric rods, as
shownin Fig. 8(a). This demonstration, combined with a rel-evant
analysis on the toroidal response of high indexdielectrics,
suggests that the most natural platform forpractical realization of
toroidal response in the THzregion is a system of polaritonic rods,
such as LiTaO3
rods, which exhibit large permittivity values even wellbelow the
phonon resonance frequency ωT – see Eq. (1).Note that for LiTaO3
ωT/2π = 26.7 THz, ωL/2π = 46.9,�/2π = 0.94 THz and ε∞/ε0 = 13.4
[20].
-
s – Fun
[
[
[G. von Freymann, S. Linden, M. Wegener, Gold helix
photonicmetamaterial as broadband circular polarizer, Science 325
(2009)1513–1515.
[13] S. O’Brien, J.B. Pendry, Photonic band-gap effects and
magnetic
384 M. Kafesaki et al. / Photonics and Nanostructure
At frequencies of few THz, which lie well below thephonon
resonance frequency ωT, the relative permittiv-ity of LiTaO3 is
still high, reaching ε = 41.4 and allowingthus the demonstration of
toroidal response in systemslike the one of Fig. 8(a), where the
meta-molecule iscomposed of 4 closely placed and aligned
cylinders.In Fig. 8(b) we demonstrate the toroidal response insuch
a system (made of by LiTaO3 rods) by plottingthe magnetic field and
the resulting magnetic flux inthe meta-molecule shown in Fig. 8(a).
The torus-likeshape of the magnetic flux is a demonstration of
thetoroidal response which was further confirmed by com-paring
simulations data with results of analysis based onmultiplole
expansion of scattered fields [71].
7. Conclusions
Materials exhibiting a phonon-polariton resonance(polaritonic
materials) offer unique possibilities inmetamaterial-based
phenomena and applications, as aresult of the rich permittivity
response that they exhibit,ranging from large positive to large
negative and nearzero values, combined with their THz operation
regime,where most of the natural materials do not show
strongresponse.
Here we discussed metamaterials composed of cylin-drical rods in
a host where both rods and host aremade of a polaritonic material.
We demonstrated numer-ically a variety of phenomena and
possibilities in suchmetamaterials, including hyperbolic dispersion
relationresponse, suitable for superlensing, backward
radiation,also suitable for superlensing, total transmission
andsubwavelength guiding exploiting Mie-resonances ofthe rods at
the ENZ response of the host, as well astoroidal response. Many
more possibilities can be real-ized and demonstrated at the same
system, includingnegative permeability and/or permittivity, and
even neg-ative refractive index, controlled absorption,
high-fieldconfinement appropriate for biomolecular sensing,
etc.
The study presented here was inspired and promptedmainly by the
easy practical realization of the above men-tioned metamaterials,
by employing self-organizationof alkali-halide eutectic systems.
This approach is fast,cost-effective, and allows adjustment of the
length scaleof the structure at the time of fabrication, offering
exten-sive possibilities in shapes, sizes and materials
involved.
The observation of such a rich variety of phenomenaas the ones
mentioned above in only one particular
geometrical configuration is a clear indication for thehuge
possibilities offered by polaritonic materials inconnection to
metamaterials. This is especially true inview of polaritonic
material’s operation in the THz
[
damentals and Applications 12 (2014) 376–386
region, where, in spite of the needs, there is a signifi-cant
gap in observed phenomena, appropriate materials,and devices. THz
polaritonic metamaterials seem capa-ble to fill-in the gap, to
stimulate new devices, and toopen the path for unique potential
applications.
Acknowledgements
Authors acknowledge financial support by EU projectENSEMBLE
(Grant Agreement No. 213669) and byGreek GSRT through the project
ERC-02 EXEL. Workat Ames Laboratory was partially supported by the
U.S.Department of Energy (Basic Energy Sciences, Divisionof
Materials Sciences and Engineering), Contract No.DE-AC02-07CH11358,
and by the US Office of NavalResearch (Award No.
N00014-10-1-0925).
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THz metamaterials made of phonon-polariton materials1
Introduction2 Basic systems and their electromagnetic properties3
Hyperbolic dispersion relation in THz polaritonic metamaterials4
THz superlensing based on backward radiation in negative
permittivity waveguides5 Epsilon near zero based total transmission
and subwavelength guiding in polaritonic systems6 Toroidal response
in eutectic metamaterials7
ConclusionsAcknowledgementsReferences